φ—归纳集范畴与偏序集的一个分类定理

本文引进了-归纳集范畴IP,证明了当Φ函子是并完备时,IP 与偏序集范畴 P 是等价的,并且利用 Galois 连接序列,给出偏序集的一个分类定理,它是文[4]中的分类定理在是任意并完备的Φ-函子情形下的推广.     

模糊偏序集上理想完备的幂等性

主要讨论模糊偏序集上理想完备性的本质.并得到以下结论:模糊偏序集的理想完备是幂等的当且仅当理想完备上的广义Scott拓扑与Alexandroff拓扑是一致的.     

Frink拟代数偏序集

引入了Frink拟代数偏序集的概念,讨论了它的一些性质,给出了它的若干刻画,特别地,证明了Frink拟代数偏序集的正规完备化为拟代数格。     

S-定向完备偏序集范畴

给出定向完备偏序半群的定义,研究定向完备偏序半群在定向完备偏序集上的作用.探讨S-定向完备偏序集范畴的一些基本性质,并且证明以S-定向完备偏序集为对象,以S-Scott连续映射为态射的范畴是笛卡尔闭范畴.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

偏序集上一致连续性的等价刻画与性质

将一致小于关系移植到一般偏序集上,同时引入了上界小于关系,定义了偏序集的一致连续性和上界连续性.给出了一致连续偏序集的等价刻画,探讨了一致连续偏序集所具有的性质.主要结果有:(1)证明了偏序集上的一致连续性,上界连续性与s-超连续性均等价;(2)在交半格条件下,偏序集的一致连续性等价于它的每一主理想一致连续;(3)在并半格条件下,偏序集的一致连续性蕴含连续性,反之不成立;(4)一致完备的一致连续偏序集均是连续bc-dcpo,且每个主理想均为完全分配格;(5)在一致完备的条件下,一致连续性对主滤子,对闭区间,对Scott S-集以及对一致连续投射像均是可遗传的.文中也构造了若干实用的反例.     

φ-归纳集范畴与偏序集的一个分类定理

本文引进了φ-归纳集范畴φIP,证明了当Φ西函子是并完备时,φIP与偏序集范畴P是等价的,并且利用Galois连接序列,给出偏序集的一个分类定理,它是文[4]中的分类定理在φ是任意并完备的Φ-函子情形下的推广。     

H-代数偏序集

本文引入了H-代数偏序集的概念,讨论了它的一些基本性质.得到如下主要结果:(1)举例说明了代数偏序集未必是H-代数偏序集;(2)偏序集是H-代数偏序集当且仅当强紧元是它的强基;(3)偏序集是H-代数偏序集当且仅当它的局部Scott拓扑是强代数格.     

条件交半格中的相对极大滤子

引入偏序集的相对极大滤子的概念,证明在任意条件交半格中一个滤子是相对极大滤子当且仅当它是滤子格的完全交不可约元.一个格是分配的当且仅当每一个相对极大滤子都是素滤子.随后研究了Heyting代数中相对极大滤子的刻画,最后定义和研究了完全并既约生成格.     

模糊偏序集上的扩张定理

借助于模糊Galois联络,在模糊偏序集上定义了完备扩张,并建立了模糊dcop和完备扩张之间的等价关系。此外,还定义了连续扩张,并得到模糊domain和连续扩张的等价刻画定理。     

Interval dimension and MacNeille completion

Equality between the interval dimensions of a poset and its MacNeille completion, announced in [7], has been obtained by the authors as a byproduct of their study of Galois lattices in [8]. The purpose of this note is to give a direct proof, similar to the classical proof of Baker's result stating that the dimension (in the Dushnik-Miller sense) of a poset and its MacNeille completion are the same.Supported by French PRC Math-Info.     

连续偏序集及其Smyth幂的几个等权定理

推广连续D om a in的权的概念到连续偏序集上,探讨连续偏序集的权、相应内蕴拓扑的权、定向完备化的权以及Sm yth幂D om a in的权间的关系。得到了几个等权定理:(1)连续偏序集的权与其上Scott拓扑、L aw son拓扑的权相等;(2)连续偏序集的权与其定向完备化的权相等;(3)无穷连续D om a in的权与其Sm yth幂D om a in的权相等;(4)有限D om a in的权小于或等于它的Sm yth幂D om a in的权。     

Order-preserving reflectors and injectivity

We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T₀-topological spaces are right-Kan injective hulls of a finite subcategory.     

The Patch Construction is Dual to Algebraic DCPO Representation

Using the parallel between the preframe and the suplattice approach to locale theory it is shown that the patch construction, as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion).     

A Note on Integral Structures in Some Locally Algebraic Representations of GL2

In the p-adic local Langlands correspondence for GL_2(Q_p), the following theorem of Berger and Breuil has played an important role: the locally algebraic representations of GL_2(Q_p) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither(ψ, Γ)-module techniques nor global methods. When F is a finite extension of Q_p, we also get a simpler proof of a theorem of Vignéras for the existence of integral structures for(locally algebraic) special series and for(smooth) tamely ramified principal series.     

Isomonodromic differential equations and differential categories

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between the Gauss–Manin connection and parameterized differential Galois groups.     

Order one equations with the Painlevé property

Differential equations with the Painleve property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p.     

CONNECTEDNESS,DISCONNECTEDNESS AND CLOSURE OPERATORS: FURTHER RESULTS

Abstract

Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.     

Lie symmetries and differential Galois groups of linear equations

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.